without reference to . The Bayesian approach evaluates as an alternative to , and finds the first to be in better agreement with the observations. This is because the latter hypothesis is much more diffuse, as can be anywhere in , which results in it having a very low posterior probability. To understand why, it is helpful to consider the two hypotheses as generators of the observations: Under , we choose , and ask how likely it is to see 49,581 boys in 98,451 births. Under , we choose randomly from anywhere within 0 to 1, and ask the same question. Most of the possible values for under are very poorly supported by the observations. In essence, the apparent disagreement between the methods is not a disagreement at all, but rather two different statements about how the hypotheses relate to the data: The frequentist finds that is a poor explanation for the observation. The Bayesian finds that is a far better explanation for the observation than . The ratio of the sex of newborns is improbably 50/50 male/female, according to the frequentist test. Yet 50/50 is a better approximation than most, but not all, other ratios. The hypothesis would have fit the observation much better than almost all other ratios, including . For example, this choice of hypotheses and prior probabilities implies the statement: "if > 0.49 and < 0.51, then the prior probability of being exactly 0.5 is 0.50/0.51 98%." Given such a strong preference for , it is easy to see why the Bayesian approach favors in the face of , even though the observed value of lies away from 0.5. The deviation of over 2 sigma from is considered significant in the frequentist approach, but its significance is overruled by the prior in the Bayesian approach. Looking at it another way, we can see that the prior distribution is essentially flat with a delta function at . Clearly this is dubious. In fact if you were to picture real numbers as being continuous, then it would be more logical to assume that it would impossible for any given number to be exactly the parameter value, i.e., we should assume . A more realistic distribution for in the alternative hypothesis produces a less surprising result for the posterior of . For example, if we replace with , i.e., the maximum likelihood estimate for , the posterior probability of